The "Intensity" Countoscope: Measuring particle dynamics in real space from microscopy images

Abstract

Advances in intensity-based microscopy techniques have improved our ability to quantify particle motion at microscopic scales, enabling insight into diffusion and collective dynamics. Building on this foundation, we introduce a novel real-space approach that analyses intensity fluctuations within virtual observation boxes of variable size on microscopy images. By correlating these signals, we uncover distinct temporal regimes in the mean square changes of intensity, $\langle ΔI^2(t) \rangle$, which are strongly dependent on the box size compared to the particle width. For small boxes or short timescales, $\langle ΔI^2(t) \rangle$ scales with the mean-square displacement, while for longer timescales and larger boxes, it scales with its square root. We develop a general theoretical framework that captures these regimes and explicitly apply it to a dilute colloidal suspension imaged with confocal microscopy as an experimental model system. This allows for a robust extraction of diffusion coefficients and physical insights into particle dynamics. Our method complements intensity-based and real-space analysis, offering a tool for studying individual and potentially collective behaviour directly from image intensities, even in systems where individual particles cannot be resolved.

Figures (3)

• Figure 1: The intensity countoscope to probe dynamics.(a) Part of a microscopy image of quasi-2D diffusing fluorescent particles, overlayed with an observation box of size $L = 16~\unit{\micro \metre}$. (b) Temporal fluctuations of the particle number (orange) and total intensity (blue), rescaled by the average particle intensity $I_0$, in the box in (a). (c) Mean-squared change in intensity for experiments (points), theory (solid lines, Eq. \ref{['eq:I_Gaussian_Square']}), and in the long time limit (dashed lines, Eq. \ref{['eq:msi_infty_theo']}). Colours, from dark to light, indicate different box sizes $L$ ranging from $0.91~\unit{\micro \metre}$ (corresponding to one pixel) to $48~\unit{\micro \metre}$. (d) Intensity profile of a particle cross-section: experimental values (dots) and Gaussian fit (line), $I_0 \exp(-x^2/2\sigma^2)/(2\pi\sigma^2)$, where $\sigma$ and $I_0$, are determined from the fit in (e). (e) Intensity variance rescaled with mean intensity dependent on the observation box size for experiments (dots) and theory (line). Blue colours for the dots are chosen according to the box size as in (c).
• Figure 2: Different rescalings of intensity fluctuations $\left< \Delta I(t)^2\right>$ reveal different physical phenomena. (a) Case $\sigma \ll L$, where $\left< \Delta I(t)^2\right>$ is rescaled by the average intensity in a box $\left<I\right>$ and in time by the diffusive timescale $L^2/4D$. (b) Case $\sigma\gg L$, where $\left< \Delta I(t)^2\right>$ is normalized by the mean box intensity $\left<I\right>$ and the fraction of the particle the box covers $L^2/\sigma^2$, and in time by the diffusive timescale $\sigma^2/D$. Solid gray lines show limiting functions in the respective regimes. The colour scale is the same as in Fig. \ref{['fig:1']}(c), with additional theory predictions in (a) for $L = [128, 256, 512]~\unit{\micro\metre}$. The insets in (a,b) illustrate the ratio of relevant length scales, particle width $\sigma$ and box size $L$, for each regime.
• Figure 3: Intensity fluctuations capture diffusion coefficients at different image resolutions.$\langle \Delta I(t)^2\rangle$ for different resolutions of images: (a) 512$\times$512 pixel (full resolution, $l_\text{px} = 0.91~\unit{\micro\metre}$), (b) 128$\times$128 pixel ($1/16$ of the full resolution, $l_\text{px} = 3.6~\unit{\micro\metre}$), and (c) 64$\times$64 pixel (1/64 of the full resolution, $l_\text{px} = 7.28~\unit{\micro\metre}$). The data is rescaled as in Fig. \ref{['fig:rescaling']}(a). The colour scale is the same as in Fig. \ref{['fig:1']}(c). The insets show a section of the optical image with the corresponding resolution. The diffusion coefficients are determined via fitting.